\chapter{}
In Appendix B, I tested the general form(non-linear) or the gain-loss utility function $\mu(x)$. 

Specifically, $$
\mu '( - x) = \lambda (x)\mu '(x),where\;\lambda (x) > 1,x > 0,and\;\mathop {\lim }\limits_{x \to 0} \lambda (x) = 1,\mu ''(x) \le 0
$$
$\lambda '(x)$ could be greater or less than 0. 
Conclusions are similar for $\lambda '(x) \leq 0$ and $\lambda(x)=lambda$: \\
(i) 2-state case: $q^*=1$ if $p>P^*$; $q^*=0$ if $p<P^*$;\\
(ii) S-state case: there exists more than one optimal sets of $\{q_s\}$.  

All the optimal conditions below are calculated by making $
{{\partial U} \over {\partial q_s }} = 0
$, where
$$
U = \mathop \Sigma \limits_{s \in {\cal S}} q_s^{} u(Z_s^{} ) + \eta \mathop \Sigma \limits_{s \in {\cal S}} p_s^{} \mu [u(Z_s^{} ) - \mathop \Sigma \limits_{s \in {\cal S}} q_s^{} u(Z_s^{} )]
$$
First, we presents previous conclusions under linear assumption from Proposition 3 for reference:\\ 
\begin{itemize}
\item[(1).] {Linear:} \\
$\mu (x) = x\quad (x > 0),\mu (x) = \lambda x\quad (x < 0),
$
(i) 2-state case: \\
$P^*  = {{\eta \lambda  - 1} \over {\eta (\lambda  - 1)}} = 1 + {{\eta  - 1} \over {\eta (\lambda  - 1)}}$ is independent of $q$. \\
Optimal q is: $q=0$, if $p<P^*$; $q=1$, if $p>P^*$. 

(ii) S-state case:\\
Optimal $
\{ q_s^{} \} _{s \in {\cal S}} 
$ satisfies: $$
P_ +   = P^*  = {{\eta \lambda  - 1} \over {\eta (\lambda  - 1)}},where\;P_ +   = \mathop \Sigma \limits_A p_s ,A = \{ s \in {\cal S}:\>u(Z_s ) - \mathop \Sigma \limits_{s \in {\cal S}} q_s u(Z_s ) \ge 0\} 
$$, 
and $P^*$ is independent of $q_s$. Therefore, there exist more than one optimal sets of $
\{ q_s^{} \} _{s \in {\cal S}} 
$, and the only constraint for those sets to be optimal is that $
\mathop \Sigma \limits_{s \in {\cal S}} q_s u(Z_s )
$ achieves a certain value (for given$
\{ u(Z_s )\} _s^{} 
$), which is uniquely determined by $
P^*  = {{\eta \lambda  - 1} \over {\eta (\lambda  - 1)}}
$ for given $
\{ p_s^{} \} _{s \in {\cal S}} 
$. (Conclusions from Proposition 3). 

\item[(2)]{Non-linear:}\\
Assumption: $$
\mu '( - x) = \lambda (x)\mu '(x),where\;\lambda (x) > 1,x > 0,and\;\mathop {\lim }\limits_{x \to 0} \lambda (x) = 1,\mu ''(x) \le 0
$$.\\
(i) 2-state case; $u(1)=1, u(0)=0$.\\
From FOC, we have $
p = 1 + {{\eta \mu '(1 - q) - 1} \over {\eta [\mu '(q)\lambda (q) - \mu '(1 - q)]}}$\\
As $0<p<1$, ${{\eta \mu '(1 - q) - 1} \over {\eta [\mu '(q)\lambda (q) - \mu '(1 - q)]}} < 0$,  the denominator and the numerator must have opposite signs. \\
If $
\lambda '(x) \le 0
$, then the cut-off value 
$$
{{\eta \mu '(1 - q) - 1} \over {\eta [\mu '(q)\lambda (q) - \mu '(1 - q)]}} < 0
$$ 
is decreasing in q. Therefore, if $
p > 1 + {{\eta \mu '(1 - p) - 1} \over {\eta [\mu '(p)\lambda (p) - \mu '(1 - p)]}}
$, then the optimal $q^*=1$, otherwise, optimal $q^*=0$. 

To be more specific, let $
\mu '(x) = \beta 
$, then \\
$$
p = {{\eta \beta \lambda (q) - 1} \over {\eta \beta [\lambda (q) - 1]}} = 1 + {{\eta \beta  - 1} \over {\eta \beta [\lambda (q) - 1]}}
$$. 
For $0<p<1$, we must have $
\eta \beta  < 1
$. \\
If $
\lambda '(q) \le 0
$, then $
{{\eta \beta \lambda (q) - 1} \over {\eta \beta [\lambda (q) - 1]}}
$ is decreasing in q. Optimal $q^*=1$ if $
p > {{\eta \beta \lambda (p) - 1} \over {\eta \beta [\lambda (p) - 1]}}
$ and optimal $q^*=0$ otherwise. \\
(ii) S-state case:\\
From FOC, we have $$
\eta \{ \sum\limits_{Gain} {p_s^{} } \mu '(u_s^{}  - \sum\limits_S {q_s^{} u_s^{} } ) + \sum\limits_{Loss} {p_s^{} } \mu '(\sum\limits_S {q_s^{} u_s^{} }  - u_s^{} )\lambda (\sum\limits_S {q_s^{} u_s^{} }  - u_s^{} )\}  = 1
$$. 
Specifically, for $
\mu '( \cdot ) \equiv \beta 
$, we have, $
\sum\limits_{Loss} {p_s^{} } [\lambda (\sum\limits_S {q_s^{} u_s^{} }  - u_s^{} ) - 1] = {{1 - \eta \beta } \over {\eta \beta }}
$. \\
As the $LHS>0$, we have $\eta \beta<1$. \\
For $\lambda '(\cdot)\leq0$, if $
\sum\limits_{Loss} {p_s^{} } [\lambda (\sum\limits_S {q_s^{} u_s^{} }  - u_s^{} ) - 1] < {{1 - \eta \beta } \over {\eta \beta }}
$, then total utility is increasing in $
\sum\limits_S {q_s^{} u_s^{} } 
$. An increase in $
\sum\limits_S {q_s^{} u_s^{} } 
$ will decrease the value of $
\sum\limits_{Loss} {p_s^{} } [\lambda (\sum\limits_S {q_s^{} u_s^{} }  - u_s^{} ) - 1]
$(if the range of loss remains the same), making it further below $
{{1 - \eta \beta } \over {\eta \beta }}
$. Therefore, with the change of loss region, optimal sets of $\{q_s\}$ are those satisfy $$
\sum\limits_{Loss} {p_s^{} } [\lambda (\sum\limits_S {q_s^{} u_s^{} }  - u_s^{} ) - 1] = {{1 - \eta \beta } \over {\eta \beta }}
$$, 
where $
Loss = \{ s \in {\cal S}:\>u_s^{}  - \sum\limits_{s \in {\cal S}} {q_s u_s^{} }  < 0\} 
$. 
\end{itemize}





























